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Autoregressive Moving Average Model

ARMA(p,q) Model

For some observed time series, a very high-order AR or MA model is needed to model
the underlying process well. In this case, a combined autoregressive moving average
(ARMA) model can sometimes be a more parsimonious choice.

An ARMA model expresses the conditional mean of
yt as a function of both past
observations, yt−1,…,yt−p, and past innovations, εt−1,…,εt−q.The number of past observations that
yt depends on,
p, is the AR degree. The number of past innovations that
yt depends on,
q, is the MA degree. In general, these models are denoted by
ARMA(p,q).

The signs of the coefficients in the AR lag operator polynomial, ϕ(L), are opposite to the right side of Equation 1. When specifying and interpreting AR
coefficients in Econometrics
Toolbox, use the form in Equation 1.

Stationarity and Invertibility of the ARMA Model

Consider the ARMA(p,q) model in lag operator
notation,

ϕ(L)yt=c+θ(L)εt.

From this expression, you can see that

yt=μ+θ(L)ϕ(L)εt=μ+ψ(L)εt,

(3)

where

μ=c(1−ϕ1−…−ϕp)

is the unconditional mean of the process, and ψ(L) is a rational, infinite-degree lag operator polynomial, (1+ψ1L+ψ2L2+…).

Note

The Constant property of an arima model
object corresponds to c, and not the unconditional mean
μ.

By Wold’s decomposition [2], Equation 3 corresponds to a stationary stochastic process
provided the coefficients ψi are absolutely summable. This is the case when the AR polynomial, ϕ(L), is stable, meaning all its roots lie
outside the unit circle. Additionally, the process is causal
provided the MA polynomial is invertible, meaning all its
roots lie outside the unit circle.

Econometrics
Toolbox enforces stability and invertibility of ARMA processes. When you
specify an ARMA model using arima, you get an error if you enter
coefficients that do not correspond to a stable AR polynomial or invertible MA
polynomial. Similarly, estimate imposes stationarity and
invertibility constraints during estimation.